Geometrically nonlinear modelling of pre-stressed viscoelastic fibre-reinforced composites with application to arteries
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ORIGINAL PAPER
Geometrically nonlinear modelling of pre‑stressed viscoelastic fibre‑reinforced composites with application to arteries I. I. Tagiltsev1,2 · A. V. Shutov1,2 Received: 5 May 2020 / Accepted: 18 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Mechanical behaviour of pre-stressed fibre-reinforced composites is modelled in a geometrically exact setting. A general approach which includes two different reference configurations is employed: one configuration corresponds to the load-free state of the structure and another one to the stress-free state of each material particle. The applicability of the approach is demonstrated in terms of a viscoelastic material model; both the matrix and the fibre are modelled using a multiplicative split of the deformation gradient tensor; a transformation rule for initial conditions is elaborated and specified. Apart from its simplicity, an important advantage of the approach is that well-established numerical algorithms can be used for pre-stressed inelastic structures. The interrelation between the advocated approach and the widely used “opening angle” approach is clarified. A full-scale FEM simulation confirms the main predictions of the “opening angle” approach. A locking effect is discovered: in some cases the opening angle of the composite is essentially smaller than the opening angles of its individual layers. Thus, the standard cutting test typically used to analyse pre-stresses does not carry enough information and more refined experimental techniques are needed. Keywords Pre-stresses · Finite strain viscoelasticity · Fibre-reinforced composites · Cutting test · Opening angle approach · Efficient numerics List of symbols 𝐅 Deformation gradient 𝐂 Right Cauchy–Green tensor Ψ Helmholtz free energy per unit mass 𝐓̃ 2nd Piola–Kirchhoff stress tensor 𝟏 Identity tensor 𝐓 Cauchy stress tensor 𝐀T Transpose of a tensor tr(𝐀) Trace of a tensor 𝐀 Unimodular part of a tensor 𝐀D Deviatoric part of a tensor K Current configuration K̃ lf Load-free reference configuration K̃ sf Stress-free reference configuration * A. V. Shutov [email protected] I. I. Tagiltsev [email protected] 1
Lavrentyev Institute of Hydrodynamics, pr. Lavrentyeva 15, Novosibirsk, Russia 630090
Novosibirsk State University, ul. Pirogova 1, Novosibirsk, Russia 630090
2
1 Introduction Biological tissues like arteries, tendons and muscles may be considered as composites, consisting of soft isotropic matrix reinforced with embedded stiff fibres (Owen et al. 2018). Such materials sustain large cyclic strains and show viscoelastic anisotropic mechanical behaviour. In recent years many material models were introduced to take numerous mechanical phenomena into account: not only anisotropic hyperelasticity (Chuong and Fung 1983; Vaishnav et al. 1973; Holzapfel et al. 2000; Shearer 2015; Von Hoegen et al. 2018), viscoelasticity (Holzapfel and Gasser 2001; Latorre and Montáns 2015; Tagiltsev et al. 2018; Liu et al. 2019) and elasto-plasticity (
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